As Dickfore points out, the 3D Einstein equation has a completely different physical character than the 4D one. All degrees of freedom in the curvature are locally determined by the matter distribution. This means, for example, that there can be no gravitational radiation. And there can be no...
It has been known for a very long time that extremal (and superextremal) black holes are valid solutions of Einstein's equation. The issue with cosmic censorship has always been whether these things can arise "in practice." The meaning of that qualifier is imprecise and has changed several times...
Fermat principles guarantee local extrema. It should also be pointed out that gravitational lensing is not particularly special for involving local maxima. The same thing also happens when looking at the paths of ordinary objects freely falling in a gravitational field.
Anyway, the time that is...
If you add enough energy to the electrons to overcome the electrostatic repulsion, you'll be adding a significant amount of mass (as well as charge) when those electrons are absorbed. As far as anyone can tell right now, this extra mass will always be large enough that the inequality quoted by...
Nothing inside the horizon can propagate out, but we're only talking about radiation originating from things outside of the horizon. From an external perspective, things don't ever quite fall through the horizon anyway. And in practice, the evidence of things falling in that we're talking about...
I agree, but there's no distinct time where you'd look at the radiation and say "the charge was absorbed now."
The original article appears to be http://link.aip.org/link/JMAPAQ/v12/i9/p1845/s1" , but the result probably isn't too obvious from their equations. To quote,
You can also see...
No changes in the electric field are ever going to travel faster than light. It should be said, however, that the meaning of this statement is much more subtle than in flat spacetime: light may take multiple paths between two events, and some field components may propagate faster than some of...
No, those two actions are unrelated. The Einstein-Hilbert action is effectively the action only for the metric. Extra terms must be added in the presence of matter (if present).
The action you gave for a point particle is one possibility. At least formally, GR with a point particle could be...
This is done for spherical stars in most textbooks. See, e.g., section 6.2 of Wald or chapter 23 of MTW.
There's been a lot of work on the interpretation of various parameters in the cylindrically symmetric vacuum solutions to Einstein's equation. This is often done by providing a matter...
To say that the matter fields have compact support means that their intersection with a Cauchy surface has compact support in the mathematical sense. This is usually used loosely to mean that all matter fields (exactly) vanish outside of some finite spatial region. Saying that the matter fields...
I think one other confusion is that people can use the words scalar, vector, etc. to mean things that transform in the usual way under some particular class of coordinate transformations that may not be completely general.
So speed is a scalar in Newtonian mechanics with respect to any...
Speed is not a scalar in any natural sense. It's frame-dependent. Time is also not a component of a 4-vector. It is common to represent spacetime points in special relativity as if they were 4-vectors, but this is misleading. It only works in globally inertial (i.e. Minkowski) coordinates. Edit...
I think that the author is claiming to consider two point particles (without "internal structure"). This makes no sense in full GR, but it is perfectly compatible with \partial_\mu T^{\mu\nu}=0. Plugging in the relevant form for the stress-energy tensor, it is easily shown that such particles...